A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

-1
votes
1answer
27 views

Geometric Brownian Motion [on hold]

I am new there. How can I calculate following expected value: $$E[X(s)\times X(t)]$$ where $X$ is Geometric Brownian Motion, i.e. $X(t) = exp[(\mu - 0.5\cdot \sigma^2)t + \sigma\cdot W(t)]$ ...
1
vote
0answers
10 views

Application of Doob's Optional Stopping Time Theorem on new stopping time

Consider a random walk on a line starting at 0. On each step the probability of moving in either direction (right or left) is 1/2. There are two particular points on the line -a, and b. If $\tau$ is ...
1
vote
1answer
18 views

Exponential of a uniform integrable martingale is a submartingale

For reference I want to prove this Lemma: Let $M$ be a uniformly integrable martingale with the additional property that $\mathbb{E}[ \exp(M_\infty)] < 1$. Then $\exp(M)$ is a uniformly ...
0
votes
1answer
50 views

Compute Var(x=X1+X2+…+Xn)

Compute $Var(X_1+X_2+...+X_n)$ given $X_1,X_2...$ are iid.,$EX=\mu,Var(X)=\sigma ^2$,and $Var(N)=\sigma [n]^2$, N is a random variable of nonnegative integers independent with X, and my solution ...
2
votes
1answer
27 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
0
votes
0answers
16 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
0
votes
1answer
17 views

Expectation of a Wiener process at a Stopping Time - 2

I am working through an answer to the following question and I do not understand a statement given towards the end of the solution, specifically why $\tilde{W}(\sigma) = 1$. (This question is related ...
0
votes
0answers
14 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
0
votes
0answers
13 views

What is the resulting stochastic process of divided Geometric Brownian motions

Let $W_{1,t},W_{2,t},...,W_{n,t}$ be $n$ independent geometric Brownian motions. Now let's say I construct the following processes: $$ X_1 = \frac{W_1}{\sum_i^n W_{i,t}} $$ $$ X_2 = ...
0
votes
0answers
21 views

Renewal process past exam question

Consider a renewal process ($N_t$, t ≥ 0) with independent inter-occurrence times $X_n$, n ∈ N, all having the same cumulative distribution function: $P(X_1 ≤ x) = w_1*F_1(x) + w_2*F_2(x)$, $w_1, w_2 ...
1
vote
0answers
24 views

Markov factorization of the density of an AR(1) process

Suppose we have a causal, stationary AR$(1)$ process with i.i.d. innovations $Z_t$. Then we know that it is a Markov as future value $X_{t+1} = \phi X_t + Z_{t+1}$ given the past $X_1,\ldots X_t$ ...
1
vote
2answers
17 views

Random Variable being $F$-measurable

It is said the Random variable is $F$-measurable if $\{X\leq x\}$ is an element of $F$. Is $X$ not $F$measurable once it is not less than or equal to $1$ $x$ or only for all?
3
votes
2answers
48 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
0
votes
0answers
7 views

Stationary distribution of a stochastic process

I have a discrete time stochastic process $\{X_t : t \in T\}$ with continuous state space. Assume $X_0=0$ and increments $\delta_t$ are exponential with mean $\alpha$ (so its parameter is ...
0
votes
1answer
24 views

Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
0
votes
0answers
13 views

moment generating function with Taylor series simplification

Denotes $a(0,r_1,r_2)$ as the annulus with radii $r_1<r_2$ centered at the origin $0$ Consider two bands $a(0,s,t)$ and $a(0,u,\sim)$ for $1\leq s\leq t\leq u$ Suppose a variable (call it an ...
-2
votes
0answers
10 views

Multi-step probability problem. Noise and Stochastic Processes. [on hold]

Please see the image below! I am having issues with this problem and would love a solution.
1
vote
0answers
16 views

Proof: Sum of two independent gaussian vectors is a gaussian vector

I want to show that the sum of two independent gaussian vectors is a gaussian vector. We had, that a gaussian vector can be written as $X=A*Z+b$ where $A$ is a real matrix, $b$ is a real vector and ...
2
votes
1answer
43 views

Showing $E[X_{n+1}|X_1,…,X_n] = a_0+\Sigma_{k=1}^n a_kX_k$

$X_1,...,X_n,X_{n+1}$ are jointly distributed with a Gaussian distribution. We let $X^* = E[X_{n+1}|X_1,...,X_n]$. Show that there exists constants $a_1,...,a_n,a_{n+1}$ such that $X^* = ...
1
vote
0answers
15 views

Extension of Law of Iterated Logarithms

Suppose I have a stochastic differential equation ($X_t$ is a vector) $dX_t = f(X_t) dt + \sigma g(X_t) d\eta(t)$ and define $V = \sum_{i=1}^{n} x_i$. Here, $\eta(t)$ is an Ornstein-Uhlenbeck process. ...
1
vote
0answers
7 views

How to prove exchangeability for a renewal process of inter-arrival times

By definition we have that $X_1, \ldots , X_n $ are exchangeable if $X_{i_1}, \ldots, X_{i_n}$ has the same joint distribution as $X_1, \ldots , X_n $ whenever $i_1, \ldots,i_n$ is a permutation of ...
1
vote
1answer
36 views

Interchange of the expected value and infinite summation $E(\sum_{m=0}^\infty (it)^m Y_t^m/m!)=\sum_{m=0}^\infty E((it)^m Y_t^m/m!)$

Let $Y_t$ be a random variable (Not positive necesarily). Can I make the next assumption? $$E(\sum_{m=0}^\infty (it)^m Y_t^m/m!)=\sum_{m=0}^\infty E((it)^m Y_t^m/m!)$$ Thanks! I think it is correct ...
0
votes
1answer
27 views

Geometric brownian motion - Ito's lemma

I have a question about geometric brownian motion. dS = uSdt + /sigma/SdW and then we do log(S) and we want to found dlog(S). So we use Ito's lemma en I get the dt part of the lemma but I don't see ...
0
votes
0answers
8 views

Markov Process under Binomial model

I have the following definition of a markov process: Consider the Binomial asset-pricing model. Let $X_0$, $X_1$.., $X_n$ be an adapte process. If for every $n$ between $0$ and $N-1$ and for ...
1
vote
1answer
24 views

Examples of Wiener Martingales

$(X_t,\mathcal{F}_t)$ is called a Weiner martignale if i) $X_t$ is a Wiener Process ii) $(X_t,\mathcal{F}_t)$ is a martingale. (Here $\mathcal{F}_t$ is an increasing $\sigma$-field family). Let ...
2
votes
1answer
31 views

Checking if $B_t^3 $ and $3tB_t$ are martingales?

$$\mathbb{E}[ B_t^3 - 3tB_t + 3B_t | \mathcal{F}_s]$$ $$\mathbb{E}[B_t^3 | \mathcal{F}_s] - 3\mathbb{E}[t B_t | \mathcal{F}_s\}$$ $$\mathbb{E}[(B_t^3 - B_s^3 + B_s^3) | \mathcal{F}_s] + [ not \space ...
2
votes
1answer
21 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
0
votes
0answers
29 views

Jump-Diffusion Process: How to calculate the expectation of integral of S(t)

Having a jump-diffuion process $S(t)$ and the transition density $f_{dS(t)}(x)$. How can I calculate the Expectation of the integral of $S(t)$ between two instants $t_0$ and $t_1$? $S(t_0)$ is ...
5
votes
1answer
47 views

Filtrations and Sigma-Algebras and Stopping Times

In a previous post Filtrations and Sigma-Algebras I asked the question: $\textbf{Previous Question:}$ Let $\Omega=\{1,2,3\}, \mathcal{A}=\mathcal{P}(\Omega)$ and $P(\{\omega\})=\tfrac{1}{3}$ for each ...
0
votes
0answers
28 views

If there are two different stationary distributions, then there are infinitely many distributions in reducible markov chain

If there are two stationary distributions μ1 and μ2 there are actually infinitely many stationary distributions: (pμ1 + (1 − p)μ2) is also a stationary distribution for any real number 0 ≤ p ≤ 1. How ...
0
votes
0answers
17 views

Stochastic Integral martingale if no $dt$ term? [duplicate]

There is a proposition in my book that For a process $M_t$ to be a martingale, it is necessary that its stochastic differential $dM_t$ has no $dt$ term. Why is this exactly? My guess is that it ...
1
vote
1answer
8 views

Path of diffusion process with discontinuous drift

Let $(B_t)$ be a standard Brownian motion on some probability space and let $X_t$ be the process defined by the SDE $dX_t = \mu_t dt + dB_t$, where $\mu_t$ is adapted, deterministic, and only takes ...
1
vote
1answer
25 views

SDE Modeling: Ito vs. Stratonovich

In my SDE class last semester there were some hints that sometimes an SDE model makes more sense in the Ito sense, and sometimes in the Stratonovich sense. This was explained very briefly and vaguely. ...
7
votes
1answer
89 views

Show that $f(W_t)-\frac{1}{2} \int_0^t f''(W_s) \, ds$ is a martingale without using Itô's formula

I'm learning the basics about Brownian motion (I know nothing about stochastic calculus), and I've shown that if $W(t)$ is a standard Brownian motion, then $W(t)^2-t$ is a martingale. Now I'm trying ...
0
votes
0answers
13 views

Showing that $X_t = \int^{1/[X]_t}_0 f_u dW_u$ is a Brownian motion

Assume we have an Ito process $$ X_t = \int^t_0 f_u d W_u $$ where $f_u$ is a deterministic function of $u$ and $W_u$ is a Brownian motion adapted to $\lbrace \mathcal F_t \rbrace$. I want to show ...
2
votes
1answer
23 views

The finite-dimensional distributions of a centered Gaussian process are uniquely determined by the covariance function

Let $I\subseteq\mathbb{R}$ and $X=(X_t)_{t\in I}$ be a centered Gaussian process, i.e. - $E[X_t]=0$ for all $t\ge 0$ - $X$ is real-valued and for all $n\in\mathbb{N}$ and $t_1,\ldots,t_n\ge 0$ we've ...
3
votes
1answer
43 views

How to compute stochastic integral: $\int_0^t d(B_s^2)$

Here, $B_t$ is Brownian motion at time $t$ What property is used to compute the integreal $\int_0^t d(B_s^2)$? Shouldn't there be some other variable attached with the differential $d$ ?
4
votes
1answer
35 views

Deriving master equation for discrete process

Consider a group of $N$ professors, $Y$ of whom are wearing white socks and $X = N − Y$ others who are wearing black socks. On each time step, one professor is chosen at random and he has to put a new ...
0
votes
0answers
22 views

what does it mean that a system is attractive?

What does it mean that a system is attractive in the context of Statistical Mechanics? Is this notion related to the presence of some monotonicity properties?
-1
votes
1answer
22 views

Two Poisson processes [closed]

Suppose we have two independent homogeneous Poisson processes with intensities $a$ and $b$. What is the distribution the number of arrivals of the first process strictly between $k$-th and $m$-th ...
-2
votes
2answers
39 views

Homogeneous Poisson process conditional expectation [closed]

I have a homogeneous Poisson Process with intensity $\lambda$, $S_n$ time of $n$-th arrival, $N_t$ number of arrivals till time $t$. How to compute $E(S_1\mid N_1\ge 2)$?
1
vote
1answer
31 views

Is Brownian Motion increasing?

Given a process $Y_t = e^{B_t}$ We know that since Brownian motion is continuous for $t \geq 0$. Since $B_t$ is a completely random motion, it is true that we cannot say whether it is monotone ...
3
votes
1answer
46 views

How to calculate $\mathbb{E}((B_3-B_2)(B_4-B_{\pi}) \mid B_1)$ for a Brownian motion $(B_t)_{t \geq 0}$

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
3
votes
0answers
23 views

Best predictor of Brownian motion

Let $B_t$ be brownian motion at time $ t \geq 0$. Then I want to find the best predictor of $B_8 + 4$ given that there are observations of brownian motion up to time $t = 1$. Approach: Essentially, ...
2
votes
1answer
13 views

Conditional expectation and brownian motion - check my answer please

$X = \frac{ B_1+ B_3 - B_2}{\sqrt{2}}$ and $Y = \frac{B_1 - B_3+ B_2}{\sqrt{2}}$ Where $B_t$ Is brownian motion at time $t\geq0$ I want to find $\mathbb{E} [Y + 3X | X]$ It is known to me that $X, ...
0
votes
0answers
20 views

Using Feynamn-Kac to solve expectation

I want to solve an expectation $u(t,x)=\mathbb{E}[exp(\beta X_T)|\mathcal{F}_t]$ related to a random variable $X$ which statisfied the CIR process $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$, which ...
2
votes
1answer
19 views

Independence of two random variables derived from a Brownian motion

If $X = B_1 + B_3 - B_2$ and $Y = B_1 - B_3 + B_2$ Where $B_t$ is Brownian Motion for $t \geq 0$ And I want to state with certainty whether $X$ and $Y$ are indep or not, do I simply just ...
3
votes
1answer
34 views

Distribution of Brownian Motion help

If $X = \frac{B_1 - B_3 + B_2}{\sqrt{2}}$ Where $B_t$ is brownian motion at time $t$. And I want to find the the distribution of $X$, how would I do so? $E[X] = 0$ is fairly straight forward. For ...
2
votes
1answer
50 views

Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the filtration generated by $W$ and let $X$ be a process that is progressively measurable ...
0
votes
0answers
17 views

conditional expectation of a sum of two independent Ornstein-Uhlenbeck type processes

Consider two independent processes of discrete Ornstein-Uhlenbeck type, $X_t$ and $Y_t$: \begin{eqnarray*} dX_t&=&\theta_x(\bar{X}-X_t) + \sigma_xdB_{xt}\\ dY_t&=&\theta_y(\bar{Y}-Y_t) ...